Dimensions

Hermann Minkowski was a Russian born German mathematician, of Jewish and Polish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.

At the Eidgenossische Polytechnikum he was one of Einstein’s teachers. Minkowski explored the arithmetic of quadratic forms, especially concerning “n” variables, and his research into that topic led him to consider certain geometric properties in a space of multiple dimensions. In 1896, he presented his geometry of numbers, a geometrical method that solved problems in number theory.

By 1907 Minkowski realized that the special theory of relativity, introduced by Einstein in 1905, could be best understood in a four dimensional space, since known as Minkowski Spacetime, in which the time and space are not separated entities but intermingled in a four dimensional space time, and in which the geometry of special relativity can be nicely represented.

In physics, spacetime is any mathematical model that combines space and time into a single construct called the spacetime continuum. Spacetime is usually interpreted with space being three dimensional and time playing the role of the fourth dimension. According to Euclidean space perception, the universe has three dimensions of space and one dimension of time. By combining space and time into a single manifold, physicists have significantly simplified a large number of physical theories, as well as described in a more uniform way the workings of the universe at both the supergalactic and subatomic levels.

The concept of spacetime combines space and time within a single coordinate system, typically with four dimensions: length, width, height, and time. Dimensions are components of a coordinate grid typically used to locate a point in space, or on the globe, such as by latitude, longitude and planet (Earth). However, with spacetime, the coordinate grid is used to locate events rather than just points in space, so time is added as another dimension to the grid.

Formerly, from experiments at slow speeds, time was believed to be a constant, which progressed at a fixed rate. However, later high speed experiments revealed that time slowed down at higher speeds with such slowing called time dilation. Many experiments have confirmed the slowing from time dilation, such as atomic clocks onboard a Space Shuttle running slower than synchronized Earth clocks. Since time varies, it is treated as a variable within the spacetime coordinate grid, and time is no longer assumed to be a constant, independent of the location in space.

Treating spacetime events with the four dimensions which include time is the conventional view. However, other invented coordinate grids treat time as three additional dimensions, with length time, width time, and height time, to accompany the three dimensions of space. When dimensions are understood as mere components of the grid system rather than physical attributes of space, it is easier to understand the alternate dimensional views, such as latitude, longitude, plus Greenwich Mean Time (three dimensions), or city, state, postal code, country, and UTC time (five dimensions). The various dimensions are chosen, depending on the coordinate grid used.

The term spacetime has taken on a generalized meaning with the advent of higher-dimensional theories. How many dimensions are needed to describe the universe is still an open question. Speculative theories such as string theory predict 10 or 26 dimensions with some predicting 11 dimensions consisting of 10 spatial and 1 temporal, but the existence of more than four dimensions would only appear to make a difference at the subatomic level.

In theoretical physics, Minkowski space is often compared to Euclidean space. While a Euclidean space has only spacelike dimensions, a Minkowski space has also one timelike dimension. Therefore the symmetry group of a Euclidean space is the Euclidean group and for a Minkowski space it is the Poincare group.

The beginning part of his address delivered at the 80th Assembly of German Natural Scientists and Physicians in 1908 is now famous:

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

Strictly speaking, the use of the Minkowski space to describe physical systems over finite distances applies only in the Newtonian limit of systems without significant gravitation. In the case of significant gravitation, spacetime becomes curved and one must abandon special relativity in favor of the full theory of general relativity.

Even in such cases, Minkowski space is still a good description in an infinitesimally small region surrounding any point, barring gravitational singularities. More abstractly, we say that in the presence of gravity spacetime is described by a curved four dimensional manifold for which the tangent space to any point is a four dimensional Minkowski space. Thus, the structure of Minkowski space is still essential in the description of general relativity.

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